Revisiting the S-matrix approach to the open superstring low energy eective lagrangian
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- Θάνος Γαλάνης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Revisiting the S-matrix approach to the open superstring low energy eective lagrangian IX Simposio Latino Americano de Altas Energías Memorial da América Latina, São Paulo. Diciembre de L. A. Barreiro (UNESP, Rio Claro - Brasil) R. Medina (UNIFEI, Itajubá - Brasil) 1
2 Revisiting the S-matrix approach to the open superstring low energy eective lagrangian JHEP10 (2012) 108 arxiv: L. A. Barreiro (UNESP, Rio Claro - Brasil) R. Medina (UNIFEI, Itajubá - Brasil) 2
3 Revisiting the S-matrix approach to the open superstring low energy eective lagrangian JHEP10 (2012) 108 arxiv: (... and work in progress...) L. A. Barreiro (UNESP, Rio Claro - Brasil) R. Medina (UNIFEI, Itajubá - Brasil) 3
4 A. (Extremely) Brief review of the tree level open string scattering amplitudes 4
5 (Tree level) N-point amplitude of gauge bosons 5
6 (Tree level) N-point amplitude of gauge bosons A N = i(2π) D δ D (k k N ) [ tr(λ a 1 λ a 2... λ a N ) A(1, 2,..., N) + ( non-cyclic permutations ) ] 6
7 (Tree level) N-point amplitude of gauge bosons A N = i(2π) D δ D (k k N ) [ tr(λ a 1 λ a 2... λ a N ) A(1, 2,..., N) + ( non-cyclic permutations ) ] A(1, 2,..., N) = g N 2 dµ(z) < ˆV 1 (z 1, k 1 ) ˆV 2 (z 2, k 2 )... ˆV N (z N, k N ) >, 7
8 (Tree level) N-point amplitude of gauge bosons A N = i(2π) D δ D (k k N ) [ tr(λ a 1 λ a 2... λ a N ) A(1, 2,..., N) + ( non-cyclic permutations ) ] A(1, 2,..., N) = g N 2 dµ(z) < ˆV 1 (z 1, k 1 ) ˆV 2 (z 2, k 2 )... ˆV N (z N, k N ) >, ˆV j (z j, k j ) Open string vertex operator. 8
9 For the BOSONIC open string: A(1, 2,..., N) = 1 xn 2 g N 2 dx N 2 dx N exp N i<j x3 0 dx 2 2α ζ i ζ j (x j x i ) N 2 i j N i<j (x j x i ) 2α k i k j 2α k j ζ i (x j x i ) linear 9
10 For the BOSONIC open string: A(1, 2,..., N) = 1 xn 2 g N 2 dx N 2 dx N exp N i<j x3 0 dx 2 2α ζ i ζ j (x j x i ) N 2 i j N i<j (x j x i ) 2α k i k j 2α k j ζ i (x j x i ) linear For the SUPERSYMMETRIC open string: A(1, 2,..., N) = 2 gn 2 (2α ) 2(x N 1 x 1 )(x N x 1 ) exp xn 1 0 N i j xn 2 dx N 2 dx N 3... dθ 1... dθ N 2 0 N i<j x3 0 dx 2 (x j x i θ j θ i ) 2α k i k j dφ 1... dφ N (2α ) 1 (θ j θ i )φ j (ζ j k i ) 1/2 (2α ) 1 φ j φ i (ζ j ζ i ) x j x i θ j θ i 10
11 Example 1 : 3-point amplitude 11
12 Example 1 : 3-point amplitude i) Open BOSONIC string A b (1, [ 2, 3) = 2g (ζ 1 k 2 )(ζ 2 ζ 3 ) + (ζ 2 k 3 )(ζ 3 ζ 1 ) + (ζ 3 k 1 )(ζ 1 ζ 2 ) ] + + 2g (2α ) (ζ 1 k 2 )(ζ 2 k 3 )(ζ 3 k 1 ) 12
13 Example 1 : 3-point amplitude i) Open BOSONIC string A b (1, [ 2, 3) = 2g (ζ 1 k 2 )(ζ 2 ζ 3 ) + (ζ 2 k 3 )(ζ 3 ζ 1 ) + (ζ 3 k 1 )(ζ 1 ζ 2 ) ] + + 2g (2α ) (ζ 1 k 2 )(ζ 2 k 3 )(ζ 3 k 1 ) ii) Open SUPERSYMMETRIC string A s (1, [ 2, 3) = 2g (ζ 1 k 2 )(ζ 2 ζ 3 ) + (ζ 2 k 3 )(ζ 3 ζ 1 ) + (ζ 3 k 1 )(ζ 1 ζ 2 ) ] 13
14 Example 2 : 4-point amplitude 14
15 Example 2 : 4-point amplitude i) Open BOSONIC string A b (1, 2, 3, 4) = 8 g 2 α 2 Γ( 1 α s)γ( 1 α t) Γ(2 α s α t) K b (ζ 1, k 1 ; ζ 2, k 2 ; ζ 3, k 3 ; ζ 4, k 4 ; α ) 15
16 Example 2 : 4-point amplitude i) Open BOSONIC string A b (1, 2, 3, 4) = 8 g 2 α 2 Γ( 1 α s)γ( 1 α t) Γ(2 α s α t) K b (ζ 1, k 1 ; ζ 2, k 2 ; ζ 3, k 3 ; ζ 4, k 4 ; α ) ii) Open SUPERSYMMETRIC string A s (1, 2, 3, 4) = 8 g 2 α 2Γ( α s)γ( α t) Γ(1 α s α t) K s (ζ 1, k 1 ; ζ 2, k 2 ; ζ 3, k 3 ; ζ 4, k 4 ), 16
17 where s = (k 1 + k 2 ) 2, t = (k 1 + k 4 ) 2, u = (k 1 + k 3 ) 2 are the Mandelstam variables 17
18 where s = (k 1 + k 2 ) 2, t = (k 1 + k 4 ) 2, u = (k 1 + k 3 ) 2 are the Mandelstam variables and K s = 1 4 [ ts(ζ 1 ζ 3 )(ζ 2 ζ 4 ) + ] +su(ζ 2 ζ 3 )(ζ 1 ζ 4 ) + ut(ζ 1 ζ 2 )(ζ 3 ζ 4 ) [(ζ 2 s 1 k 4 )(ζ 3 k 2 )(ζ 2 ζ 4 ) + (ζ 2 k 3 )(ζ 4 k 1 )(ζ 1 ζ 3 ) + ] +(ζ 1 k 3 )(ζ 4 k 2 )(ζ 2 ζ 3 ) + (ζ 2 k 4 )(ζ 3 k 1 )(ζ 1 ζ 4 ) [(ζ 2 t 2 k 1 )(ζ 4 k 3 )(ζ 3 ζ 1 ) + (ζ 3 k 4 )(ζ 1 k 2 )(ζ 2 ζ 4 ) + ] +(ζ 2 k 4 )(ζ 1 k 3 )(ζ 3 ζ 4 ) + (ζ 3 k 1 )(ζ 4 k 2 )(ζ 2 ζ 1 ) [(ζ 2 u 1 k 2 )(ζ 4 k 3 )(ζ 3 ζ 2 ) + (ζ 3 k 4 )(ζ 2 k 1 )(ζ 1 ζ 4 ) + ] +(ζ 1 k 4 )(ζ 2 k 3 )(ζ 3 ζ 4 ) + (ζ 3 k 2 )(ζ 4 k 1 )(ζ 1 ζ 2 ). 18
19 α expansion of the 4-point momentum factor: α 2Γ( α s)γ( α t) Γ(1 α s α t) = 19
20 α expansion of the 4-point momentum factor: α 2Γ( α s)γ( α t) Γ(1 α s α t) = 1 st π2 6 α 2 ζ(3)(s + t) α 3 π4 360 (4s2 + st + 4t 2 ) α 4 + [ ] π ζ(3) st(s + t) ζ(5)(s3 + 2s 2 t + 2st 2 + t 3 ) α 5 + [ ζ(3)2 st(s + t) 2 ] π (16s4 + 12s 3 t + 23s 2 t st t 4 ) α O(α 7 ). 20
21 B. (Extremely) Brief review of the low energy eective lagrangian 21
22 General Structure L e = 1 [ g 2 tr F 2 + (2α )F 3 + (2α ) 2 F 4 + (2α ) 3( F 5 + D 2 F 4) + +(2α ) 4( F 6 + D 2 F 5 + D 4 F 4) + O ( (2α ) 4) ] 22
23 General Structure L e = 1 [ g 2 tr F 2 + (2α )F 3 + (2α ) 2 F 4 + (2α ) 3( F 5 + D 2 F 4) + +(2α ) 4( F 6 + D 2 F 5 + D 4 F 4) + O ( (2α ) 4) ] + (fermions) 23
24 General Structure L e = 1 g 2 tr [ F 2 + (2α )F 3 + (2α ) 2 F 4 + (2α ) 3( F 5 + D 2 F 4) + +(2α ) 4( F 6 + D 2 F 5 + D 4 F 4) + O ( (2α ) 4) ] + + (fermions) i) Low energy eective lagrangian up to α 2 terms L e = 1 g 2tr [ 1 4 F µν F µν + (2α ) 1 a 1 F λ µ F ν λ F µ ν + + (2α ) 2 ( a 3 F µλ F ν λf ρ µ F νρ + a 4 F µ λ F λ ν F νρ F µρ + ) + a 5 F µν F µν F λρ F λρ + a 6 F µν F λρ F µν F λρ + ] + O((2α ) 3 ). 24
25 General Structure L e = 1 g 2 tr [ F 2 + (2α )F 3 + (2α ) 2 F 4 + (2α ) 3( F 5 + D 2 F 4) + +(2α ) 4( F 6 + D 2 F 5 + D 4 F 4) + O ( (2α ) 4) ] + + (fermions) i) Low energy eective lagrangian up to α 2 terms L e = 1 g 2tr [ 1 4 F µν F µν + (2α ) 1 a 1 F λ µ F ν λ F µ ν + + (2α ) 2 ( a 3 F µλ F ν λf ρ µ F νρ + a 4 F µ λ F λ ν F νρ F µρ + ) + a 5 F µν F µν F λρ F λρ + a 6 F µν F λρ F µν F λρ + ] + O((2α ) 3 ). 1 coecient at α 1 order : a 1 25
26 General Structure L e = 1 g 2 tr [ F 2 + (2α )F 3 + (2α ) 2 F 4 + (2α ) 3( F 5 + D 2 F 4) + +(2α ) 4( F 6 + D 2 F 5 + D 4 F 4) + O ( (2α ) 4) ] + + (fermions) i) Low energy eective lagrangian up to α 2 terms L e = 1 g 2tr [ 1 4 F µν F µν + (2α ) 1 a 1 F λ µ F ν λ F µ ν + + (2α ) 2 ( a 3 F µλ F ν λf ρ µ F νρ + a 4 F µ λ F λ ν F νρ F µρ + ) + a 5 F µν F µν F λρ F λρ + a 6 F µν F λρ F µν F λρ + ] + O((2α ) 3 ). 1 coecient at α 1 order : a 1 4 coecients at α 2 order: a 3, a 4, a 5, a 6. 26
27 ii) Low energy eective lagrangian at α 3 order [ tr a 10 F ν µ Fν λ L e (3) = (2α ) 3 F ρ λ F ρ σ F σ µ + a 12 Fµ ν Fν λ F σ µ F ρ λ F ρ σ + a 14 F ν µ F λ ν F µ λ F ρ σ g 2 + a 11 F ν µ Fν λ + a 13 F ν F ρ λ F µ σ F σ ρ + F λ ν F µ σ F ρ λ µ Fρ σ µ Fν λ Fρ σ F µ λ F σ ρ + ρ + Fσ ρ + a 15 F ν ( ) + a 16 Dµ Fν λ (D µ F ρ λ ) F σ ν F σ ( ) + a 17 Dµ Fν λ F ν σ (D µ F ρ λ ) F ρ σ + ( ) + a 18 Dµ Fν λ (D µ Fλ ν ) Fρ σ Fσ ρ + ( ) + a 19 Dµ Fν λ F σ ρ (D µ Fλ ν Fσ ρ + ( ) + a 20 Dσ Fµ ν F ρ ( ) λ D µ Fν λ F σ ρ + ( ) + a 21 Fµ ν D µ Fν λ F σ ρ (D σ F ρ λ ) + + a 22 Fµ ν (D µ F ρ λ ) ( ) ] D σ Fν λ F σ ρ 27
28 ii) Low energy eective lagrangian at α 3 order [ tr a 10 F ν µ Fν λ L e (3) = (2α ) 3 F ρ λ F ρ σ F σ µ + a 12 Fµ ν Fν λ F σ µ F ρ λ F ρ σ + a 14 F ν µ F λ ν F µ λ F ρ σ g 2 + a 11 F ν µ Fν λ + a 13 F ν F ρ λ F µ σ F σ ρ + F λ ν F µ σ F ρ λ µ Fρ σ µ Fν λ Fρ σ F µ λ F σ ρ + ρ + Fσ ρ + a 15 F ν ( ) + a 16 Dµ Fν λ (D µ F ρ λ ) F σ ν F σ ( ) + a 17 Dµ Fν λ F ν σ (D µ F ρ λ ) F ρ σ + ( ) + a 18 Dµ Fν λ (D µ Fλ ν ) Fρ σ Fσ ρ + ( ) + a 19 Dµ Fν λ F σ ρ (D µ Fλ ν Fσ ρ + ( ) + a 20 Dσ Fµ ν F ρ ( ) λ D µ Fν λ F σ ρ + ( ) + a 21 Fµ ν D µ Fν λ F σ ρ (D σ F ρ λ ) + + a 22 Fµ ν (D µ F ρ λ ) ( ) ] D σ Fν λ F σ ρ 13 coecients at α 3 order 28
29 C. Basics of the S-matrix approach to the (tree level) open string low energy eective lagrangian 29
30 Matching the open string amplitudes with the ones from the LEL: 30
31 Matching the open string amplitudes with the ones from the LEL: i) Case of the 3-point amplitude: Bosonic string: a 1 = i/3. Supersymmetric string: a 1 = 0. 31
32 Matching the open string amplitudes with the ones from the LEL: i) Case of the 3-point amplitude: Bosonic string: a 1 = i/3. Supersymmetric string: a 1 = 0. ii) Case of the 4-point amplitude: Coecient Bosonic Supersymmetric open string theory open string theory a 3 π 2 /12 π 2 /12 a 4 π 2 /24 π 2 /24 a 5 π 2 /48 1/8 π 2 /48 a 6 π 2 /96 + 1/8 π 2 /96 32
33 Matching the open string amplitudes with the ones from the LEL: i) Case of the 3-point amplitude: Bosonic string: a 1 = i/3. Supersymmetric string: a 1 = 0. ii) Case of the 4-point amplitude: Coecient Bosonic Supersymmetric open string theory open string theory a 3 π 2 /12 π 2 /12 a 4 π 2 /24 π 2 /24 a 5 π 2 /48 1/8 π 2 /48 a 6 π 2 /96 + 1/8 π 2 /96 It is exactly 33
34 Matching the open string amplitudes with the ones from the LEL: i) Case of the 3-point amplitude: Bosonic string: a 1 = i/3. Supersymmetric string: a 1 = 0. ii) Case of the 4-point amplitude: Coecient Bosonic Supersymmetric open string theory open string theory a 3 π 2 /12 π 2 /12 a 4 π 2 /24 π 2 /24 a 5 π 2 /48 1/8 π 2 /48 a 6 π 2 /96 + 1/8 π 2 /96 It is exactly the SAME method 34
35 Matching the open string amplitudes with the ones from the LEL: i) Case of the 3-point amplitude: Bosonic string: a 1 = i/3. Supersymmetric string: a 1 = 0. ii) Case of the 4-point amplitude: Coecient Bosonic Supersymmetric open string theory open string theory a 3 π 2 /12 π 2 /12 a 4 π 2 /24 π 2 /24 a 5 π 2 /48 1/8 π 2 /48 a 6 π 2 /96 + 1/8 π 2 /96 It is exactly the SAME method applied to two dierent expressions for A(1, 2,..., N). 35
36 D. Our revisited S-matrix approach to the (tree level) open string low energy eective lagrangian 36
37 Main observation 37
38 Main observation In the case of the superstring 38
39 Main observation In the case of the superstring A(1,..., N) does not contain (ζ k) N terms 39
40 Main observation In the case of the superstring A(1,..., N) does not contain (ζ k) N terms This is probably due to D=10 Supersymmetry 40
41 i) α 2 calculation 41
42 i) α 2 calculation a 3 = 8 a 6, a 4 = 4 a 6, a 5 = 2 a 6. 42
43 i) α 2 calculation a 3 = 8 a 6, a 4 = 4 a 6, a 5 = 2 a 6. L e = 1 g 2tr [ 1 4 F µν F µν + a 6 (2α ) 2 ( 8 F µλ F ν λf ρ µ F νρ ) 4 F µ λ F ν λ F νρ F µρ + 2 F µν F µν F λρ F λρ + F µν F λρ F µν F λρ + ] +O((2α ) 3 ). 43
44 i) α 2 calculation a 3 = 8 a 6, a 4 = 4 a 6, a 5 = 2 a 6. L e = 1 g 2tr [ 1 4 F µν F µν + a 6 (2α ) 2 ( 8 F µλ F ν λf ρ µ F νρ ) 4 F µ λ F ν λ F νρ F µρ + 2 F µν F µν F λρ F λρ + F µν F λρ F µν F λρ + ] +O((2α ) 3 ). At α 2 order there is only 1 coecient: a6. 44
45 i) α 2 calculation a 3 = 8 a 6, a 4 = 4 a 6, a 5 = 2 a 6. L e = 1 g 2tr [ 1 4 F µν F µν + a 6 (2α ) 2 ( 8 F µλ F ν λf ρ µ F νρ ) 4 F µ λ F ν λ F νρ F µρ + 2 F µν F µν F λρ F λρ + F µν F λρ F µν F λρ + ] +O((2α ) 3 ). At α 2 order there is only 1 coecient: a6. a 6 = π
46 ii) α 3 calculation 46
47 ii) α 3 calculation 2a 16 = 2a 17 = 8a 19 = a 20 = a 22, a 18 = a 21 = 0. a 11 = a 13 = 2a 15 = i a 22, a 10 = a 12 = a 14 = 0. 47
48 ii) α 3 calculation 2a 16 = 2a 17 = 8a 19 = a 20 = a 22, a 18 = a 21 = 0. a 11 = a 13 = 2a 15 = i a 22, a 10 = a 12 = a 14 = 0. L (3) e = (2α ) 3 a 22 [ g 2 tr i Fµ ν Fν λ F ρ λ F σ µ Fρ σ + i F ν i 2 F ν µ Fν λ ( ) Dµ Fν λ F ν σ + ( ) D σ Fµ ν F ρ λ F σ ρ F µ λ F ρ σ (D µ F ρ λ ) F σ ( ) D µ Fν λ F σ ρ ρ 1 8 µ Fρ σ F λ ν F µ σ F ρ λ ( ) Dµ Fν λ (D µ F ρ λ ) F σ ν Fρ σ ( ) Dµ Fν λ F σ ρ (D µ Fλ ν ) Fσ ρ Fµ ν (D µ F ρ λ ) ( ) D σ Fν λ F σ ρ ] 48
49 ii) α 3 calculation 2a 16 = 2a 17 = 8a 19 = a 20 = a 22, a 18 = a 21 = 0. a 11 = a 13 = 2a 15 = i a 22, a 10 = a 12 = a 14 = 0. L (3) e = (2α ) 3 a 22 [ g 2 tr i Fµ ν Fν λ F ρ λ F σ µ Fρ σ + i F ν i 2 F ν µ Fν λ ( ) Dµ Fν λ F ν σ + ( ) D σ Fµ ν F ρ λ F σ ρ F µ λ F ρ σ (D µ F ρ λ ) F σ ( ) D µ Fν λ F σ ρ ρ 1 8 µ Fρ σ F λ ν F µ σ F ρ λ ( ) Dµ Fν λ (D µ F ρ λ ) F σ ν Fρ σ ( ) Dµ Fν λ F σ ρ (D µ Fλ ν ) Fσ ρ Fµ ν (D µ F ρ λ ) ( ) D σ Fν λ F σ ρ At α 3 order there is only 1 coecient: a22. ] 49
50 ii) α 3 calculation 2a 16 = 2a 17 = 8a 19 = a 20 = a 22, a 18 = a 21 = 0. a 11 = a 13 = 2a 15 = i a 22, a 10 = a 12 = a 14 = 0. L (3) e = (2α ) 3 a 22 [ g 2 tr i Fµ ν Fν λ F ρ λ F σ µ Fρ σ + i F ν i 2 F ν µ Fν λ ( ) Dµ Fν λ F ν σ + ( ) D σ Fµ ν F ρ λ F σ ρ F µ λ F ρ σ (D µ F ρ λ ) F σ ( ) D µ Fν λ F σ ρ ρ 1 8 µ Fρ σ F λ ν F µ σ F ρ λ ( ) Dµ Fν λ (D µ F ρ λ ) F σ ν Fρ σ ( ) Dµ Fν λ F σ ρ (D µ Fλ ν ) Fσ ρ Fµ ν (D µ F ρ λ ) ( ) D σ Fν λ F σ ρ At α 3 order there is only 1 coecient: a22. ] a 22 = 2ζ(3) 50
51 iii) α 4 calculation: 51
52 iii) α 4 calculation: L e (4) = (2α ) 4 π 4 g 2 ( LF 6 + L D 2 F 5 + L D 4 F 4 ), where 52
53 L F 6 = t µ 1ν 1 µ 2 ν 2 µ 3 ν 3 µ 4 ν 4 µ 5 ν 5 µ 6 ν 6 (12) tr ( F µ1 ν 1 F µ2 ν 2 F µ3 ν 3 F µ4 ν 4 F µ5 ν 5 F µ6 ν 6 ), L D 2 F 5 = 56 i tµ 1ν 1 µ 2 ν 2 µ 3 ν 3 µ 4 ν 4 µ 5 ν 5 (10) tr ( F µ1 ν 1 F µ2 ν 2 F µ3 ν 3 D α F µ4 ν 4 D α F µ5 ν 5 ) + + i (η t (8)) µ 1ν 1 µ 2 ν 2 µ 3 ν 3 µ 4 ν 4 µ 5 ν 5 tr ( 169 D α F µ1 ν 1 F µ2 ν 2 F µ3 ν 3 F µ4 ν 4 D α F µ5 ν D α F µ1 ν 1 D α F µ2 ν 2 F µ3 ν 3 F µ4 ν 4 F µ5 ν F µ1 ν 1 D α F µ2 ν 2 D α F µ3 ν 3 F µ4 ν 4 F µ5 ν F µ1 ν 1 D α F µ2 ν 2 F µ3 ν 3 D α F µ4 ν 4 F µ5 ν F µ1 ν 1 F µ2 ν 2 D α F µ3 ν 3 D α F µ4 ν 4 F µ5 ν 5 + ) +16 F µ1 ν 1 F µ2 ν 2 F µ3 ν 3 D α F µ4 ν 4 D α F µ5 ν 5 i 5760 tµ 1ν 1 µ 2 ν 2 µ 3 ν 3 µ 4 ν 4 (8) { 17 ( ) tr D µ 5 F µ1 ν 1 F µ2 ν 2 F µ3 ν 3 D ν 5 F µ4 ν 4 F µ5 ν tr ( F µ1 ν 1 D µ 5 F µ2 ν 2 D ν 5 F µ3 ν 3 F µ4 ν 4 F µ5 ν 5 ) }, L D 4 F 4 = tµ 1ν 1 µ 2 ν 2 µ 3 ν 3 µ 4 ν 4 (8) tr ( D α F µ1 ν 1 D (α D β) F µ2 ν 2 D β F µ3 ν 3 F µ4 ν D α F µ1 ν 1 D α F µ2 ν 2 D β F µ3 ν 3 D β F µ4 ν 4 ). 53
54 E. Summary of our revisited S-matrix method 54
55 1. There now is an improved (revisited) version of the S-matrix approach to the Open Superstring low energy eective lagrangian, L e. 55
56 1. There now is an improved (revisited) version of the S-matrix approach to the Open Superstring low energy eective lagrangian, L e. 2. It incorporates a kinematic requiremenent in the gauge boson scattering amplitudes, (presumably) due to Supersymmetry. 56
57 1. There now is an improved (revisited) version of the S-matrix approach to the Open Superstring low energy eective lagrangian, L e. 2. It incorporates a kinematic requiremenent in the gauge boson scattering amplitudes, (presumably) due to Supersymmetry. 3. It has been successfully used to obtain the bosonic terms of L e up to α 4 order. 57
58 4. Our revisited S-matrix method suggests that, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude 58
59 4. Our revisited S-matrix method suggests that, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude are enough requirements to obtain COMPLETELY L e (order by order in α ). 59
60 4. Our revisited S-matrix method suggests that, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude are enough requirements to obtain COMPLETELY L e (order by order in α ). JHEP10 (2012)
61 5. Important remark: 61
62 5. Important remark: In the literature there already exists another method which only uses the open superstring 4-point amplitude and that succeeds in nding the α terms of L e up to α 4 order: 62
63 5. Important remark: In the literature there already exists another method which only uses the open superstring 4-point amplitude and that succeeds in nding the α terms of L e up to α 4 order: The method of BPS congurations 63
64 5. Important remark: In the literature there already exists another method which only uses the open superstring 4-point amplitude and that succeeds in nding the α terms of L e up to α 4 order: The method of BPS congurations It is due to Koerber and Sevrin ( ). 64
65 5. Important remark: In the literature there already exists another method which only uses the open superstring 4-point amplitude and that succeeds in nding the α terms of L e up to α 4 order: The method of BPS congurations It is due to Koerber and Sevrin ( ). Our results agree with the ones of Koerber and Sevrin up to α 3 order and probably also agree at α 4 order. 65
66 F.... As for the scattering amplitudes... 66
67 F.... As for the scattering amplitudes... (Work in progress) 67
68 F.... As for the scattering amplitudes... (Work in progress, to be soon sent to the hep-th ArXiv) 68
69 Interesting question: 69
70 Interesting question: Does the (tree level) N-point formula, in Open Superstring Theory, 70
71 Interesting question: Does the (tree level) N-point formula, in Open Superstring Theory, A(1, 2,..., N) = 2 gn 2 (2α ) 2(x N 1 x 1 )(x N x 1 ) exp xn 1 0 N i j xn 2 dx N 2 dx N 3... dθ 1... dθ N 2 0 N i<j x3 0 dx 2 (x j x i θ j θ i ) 2α k i k j dφ 1... dφ N (2α ) 1 (θ j θ i )φ j (ζ j k i ) 1/2 (2α ) 1 φ j φ i (ζ j ζ i ) x j x i θ j θ i 71
72 Interesting question: Does the (tree level) N-point formula, in Open Superstring Theory, A(1, 2,..., N) = 2 gn 2 (2α ) 2(x N 1 x 1 )(x N x 1 ) exp xn 1 0 N i j xn 2 dx N 2 dx N 3... dθ 1... dθ N 2 0 N i<j x3 0 dx 2 (x j x i θ j θ i ) 2α k i k j dφ 1... dφ N (2α ) 1 (θ j θ i )φ j (ζ j k i ) 1/2 (2α ) 1 φ j φ i (ζ j ζ i ) x j x i θ j θ i admit a closed expression for any N? 72
73 Interesting question: Does the (tree level) N-point formula, in Open Superstring Theory, A(1, 2,..., N) = 2 gn 2 (2α ) 2(x N 1 x 1 )(x N x 1 ) exp xn 1 0 N i j xn 2 dx N 2 dx N 3... dθ 1... dθ N 2 0 N i<j x3 0 dx 2 (x j x i θ j θ i ) 2α k i k j dφ 1... dφ N (2α ) 1 (θ j θ i )φ j (ζ j k i ) 1/2 (2α ) 1 φ j φ i (ζ j ζ i ) x j x i θ j θ i admit a closed expression for any N? Answer: 73
74 Interesting question: Does the (tree level) N-point formula, in Open Superstring Theory, A(1, 2,..., N) = 2 gn 2 (2α ) 2(x N 1 x 1 )(x N x 1 ) exp xn 1 0 N i j xn 2 dx N 2 dx N 3... dθ 1... dθ N 2 0 N i<j x3 0 dx 2 (x j x i θ j θ i ) 2α k i k j dφ 1... dφ N (2α ) 1 (θ j θ i )φ j (ζ j k i ) 1/2 (2α ) 1 φ j φ i (ζ j ζ i ) x j x i θ j θ i admit a closed expression for any N? Answer: Yes! 74
75 Interesting question: Does the (tree level) N-point formula, in Open Superstring Theory, A(1, 2,..., N) = 2 gn 2 (2α ) 2(x N 1 x 1 )(x N x 1 ) exp xn 1 0 N i j xn 2 dx N 2 dx N 3... dθ 1... dθ N 2 0 N i<j x3 0 dx 2 (x j x i θ j θ i ) 2α k i k j dφ 1... dφ N (2α ) 1 (θ j θ i )φ j (ζ j k i ) 1/2 (2α ) 1 φ j φ i (ζ j ζ i ) x j x i θ j θ i admit a closed expression for any N? Answer: Yes! Mafra, Schlotterer, Stieberger, arxiv:
76 Mafra, Schlotterer and Stieberger's result: 76
77 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), 77
78 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. 78
79 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. Examples: 79
80 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. Examples: i) N=4 : 80
81 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. Examples: i) N=4 : where A(1, 2, 3, 4) = F (4) 1 A Y M (1, 2, 3, 4), F (4) 1 = α dx 2 x 2 2α α 12 1 (1 x 2 ) 2α α 23, 81
82 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. Examples: i) N=4 : where A(1, 2, 3, 4) = F (4) 1 A Y M (1, 2, 3, 4), F (4) 1 = α dx 2 x 2 2α α 12 1 (1 x 2 ) 2α α 23, F (4) 1 = Γ(1 + 2α α 12 )Γ(1 + 2α α 23 ) Γ(1 + 2α α α α 23 ), 82
83 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. Examples: i) N=4 : where A(1, 2, 3, 4) = F (4) 1 A Y M (1, 2, 3, 4), F (4) 1 = α dx 2 x 2 2α α 12 1 (1 x 2 ) 2α α 23, F (4) 1 = Γ(1 + 2α α 12 )Γ(1 + 2α α 23 ) Γ(1 + 2α α α α 23 ), F (4) 1 = 1 (2α ) 2 ζ(2)α 12 α 23 (2α ) 3 ζ(3)α 12 α 23 (α 12 + α 23 ) + +O((2α ) 4 ) 83
84 A(1, 2, 3, 4) = F (4) 1 A Y M (1, 2, 3, 4) vs A s (1, 2, 3, 4) = 8 g 2 α 2Γ( α s)γ( α t) Γ(1 α s α t) K s (ζ 1, k 1 ; ζ 2, k 2 ; ζ 3, k 3 ; ζ 4, k 4 ), 84
85 A(1, 2, 3, 4) = F (4) 1 A Y M (1, 2, 3, 4) vs A s (1, 2, 3, 4) = 8 g 2 α 2Γ( α s)γ( α t) Γ(1 α s α t) K s (ζ 1, k 1 ; ζ 2, k 2 ; ζ 3, k 3 ; ζ 4, k 4 ), (Both expressions are equivalent) 85
86 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), where F (N) 1,..., F (N) are the momentum (N 3)! factors, which contain the α dependence. Examples: i) N=4 : where A(1, 2, 3, 4) = F (4) 1 A Y M (1, 2, 3, 4), F (4) 1 = α dx 2 x 2 2α α 12 1 (1 x 2 ) 2α α 23, F (4) 1 = Γ(1 + 2α α 12 )Γ(1 + 2α α 23 ) Γ(1 + 2α α α α 23 ), F (4) 1 = 1 (2α ) 2 ζ(2)α 12 α 23 (2α ) 3 ζ(3)α 12 α 23 (α 12 + α 23 ) + +O((2α ) 4 ) 86
87 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). 87
88 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). Examples: ii) N=5 : A(1, 2, 3, 4, 5) = = F (5) 1 A Y M (1, 2, 3, 4, 5) + F (5) 2 A Y M (1, 3, 2, 4, 5), 88
89 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). Examples: ii) N=5 : A(1, 2, 3, 4, 5) = = F (5) 1 A Y M (1, 2, 3, 4, 5) + F (5) 2 A Y M (1, 3, 2, 4, 5), where F (5) 1 = α 12 α dx 3 x3 0 dx 2 x 2 2α α 12 1 x 3 2α α 13 (1 x 2 ) 2α α 24 (1 x 3 ) 2α α 34 1 (x 3 x 2 ) 2α α 23 89
90 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). Examples: ii) N=5 : A(1, 2, 3, 4, 5) = = F (5) 1 A Y M (1, 2, 3, 4, 5) + F (5) 2 A Y M (1, 3, 2, 4, 5), where F (5) 1 = α 12 α dx 3 x3 0 dx 2 x 2 2α α 12 1 x 3 2α α 13 (1 x 2 ) 2α α 24 (1 x 3 ) 2α α 34 1 (x 3 x 2 ) 2α α 23 F (5) 1 = 1 (2α ) 2 ζ(2) ( (α 12 α 34 α 34 α 45 α 12 α 51 ) (2α ) 3 ζ(3) α12 2 α α 12 α 23 α 34 + α 12 α O((2α ) 4 ) α 2 34 α 45 α 34 α 2 45 α 2 12 α 51 α 12 α 2 51 ) 90
91 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). 91
92 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). Examples: iii) N=6 : A(1, 2, 3, 4, 5, 6) = = F (6) 1 A Y M (1, 2, 3, 4, 5, 6) + F (6) 2 A Y M (1, 3, 2, 4, 5, 6) + + F (6) 3 A Y M (1, 2, 4, 3, 5, 6) + F (6) 4 A Y M (1, 3, 4, 2, 5, 6) + + F (6) 5 A Y M (1, 4, 2, 3, 5, 6) + F (6) 6 A Y M (1, 4, 3, 2, 5, 6) 92
93 Mafra, Schlotterer and Stieberger's result: A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N). Examples: iii) N=6 : A(1, 2, 3, 4, 5, 6) = = F (6) 1 A Y M (1, 2, 3, 4, 5, 6) + F (6) 2 A Y M (1, 3, 2, 4, 5, 6) + where + F (6) 3 A Y M (1, 2, 4, 3, 5, 6) + F (6) 4 A Y M (1, 3, 4, 2, 5, 6) + + F (6) 5 A Y M (1, 4, 2, 3, 5, 6) + F (6) 6 A Y M (1, 4, 3, 2, 5, 6) F (6) 1 = α 12 α x4 x3 2α dx 4 dx 3 dx 2 x α α 2 x α x 4 2α α 14 (1 x 2 ) 2α α 25 (1 x 3 ) 2α α 35 (1 x 4 ) 2α α 45 1 (x 3 x 2 ) 2α α 23 (x 4 x 2 ) 2α α 24 (x 4 x 3 ) 2α α 34 [ α13 + α ] 23. x 3 x 3 x 2 93
94 F (6) 1 = α 12 α x4 x3 2α dx 4 dx 3 dx 2 x α α 2 x α x 4 2α α 14 (1 x 2 ) 2α α 25 (1 x 3 ) 2α α 35 (1 x 4 ) 2α α 45 1 (x 3 x 2 ) 2α α 23 (x 4 x 2 ) 2α α 24 (x 4 x 3 ) 2α α 34 [ α13 + α ] 23. x 3 x 3 x 2 94
95 F (6) 1 = α 12 α 45 1 F (6) 1 = 1 (2α ) 2 ζ(2) 0 + (2α ) 3 ζ(3) x4 x3 2α dx 4 dx 3 dx 2 x α α 2 x α 13 3 ( 0 0 x 4 2α α 14 (1 x 2 ) 2α α 25 (1 x 3 ) 2α α 35 (1 x 4 ) 2α α 45 1 (x 3 x 2 ) 2α α 23 (x 4 x 2 ) 2α α 24 (x 4 x 3 ) 2α α 34 [ α13 + α ] 23. x 3 x 3 x 2 ( α 45 α 56 + α 12 α 61 α 45 t 123 α 12 t t 123 t 345 ) + 2 α 12 α 23 α α 12 α 34 α α 2 45 α 56 + α 45 α α 2 12 α 61 + α 12 α α 34 α 45 t 123 α 2 45 t 123 α 45 t O((2α ) 4 ), 2 α 12 α 45 t 234 α12 2 t α ) 12 α 23 t t t 345 α 12 t t 123 t where α ij = k i k j and t mnp = α mn + α mp + α np. 95
96 Another interesting question: 96
97 Another interesting question: What can the revisited S-matrix method say about Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N)??? 97
98 Another interesting question: What can the revisited S-matrix method say about Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N)??? Answer: 98
99 Another interesting question: What can the revisited S-matrix method say about Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N)??? Answer: It can be re-derived from it! 99
100 Yes, 100
101 Yes, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude 101
102 Yes, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude + 102
103 Yes, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude + III. (on-shell) Gauge invariance + IV. Cyclic symmetry + V. Unitarity 103
104 Yes, demanding: I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude + III. (on-shell) Gauge invariance + IV. Cyclic symmetry + V. Unitarity Properties of gauge boson amplitudes 104
105 Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), can be re-derived. 105
106 Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), can be re-derived. At least we have succeeded in the cases of N=5 and N=6 : 106
107 Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), can be re-derived. At least we have succeeded in the cases of N=5 and N=6 (also N=7?) : 107
108 Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), can be re-derived. At least we have succeeded in the cases of N=5 and N=6 (also N=7?) : F (5) 1 = 1 (2α ) 2 ζ(2) (α 12 α 34 α 34 α 45 α 12 α 51 ) ( (2α ) 3 ζ(3) α12 2 α α 12 α 23 α 34 + α 12 α34 2 ) α34 2 α 45 α 34 α45 2 α12 2 α 51 α 12 α O((2α ) 4 ), 108
109 Mafra-Schlotterer-Stieberger's result, A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), can be re-derived. At least we have succeeded in the cases of N=5 and N=6 (also N=7?) : F (5) 1 = 1 (2α ) 2 ζ(2) (α 12 α 34 α 34 α 45 α 12 α 51 ) ( (2α ) 3 ζ(3) α12 2 α α 12 α 23 α 34 + α 12 α34 2 ) α34 2 α 45 α 34 α45 2 α12 2 α 51 α 12 α O((2α ) 4 ), F (6) 1 = 1 (2α ) 2 ζ(2) + (2α ) 3 ζ(3) + O((2α ) 4 ). ( ( α 45 α 56 + α 12 α 61 α 45 t 123 α 12 t t 123 t 345 ) + 2 α 12 α 23 α α 12 α 34 α α 2 45 α 56 + α 45 α α 2 12 α 61 + α 12 α α 34 α 45 t 123 α 2 45 t 123 α 45 t α 12 α 45 t 234 α12 2 t α 12 α 23 t ) + t t 345 α 12 t t 123 t
110 G. Global summary 110
111 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: 111
112 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. 112
113 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. L e = 1 [ g 2 tr F 2 + (2α )F 3 + (2α ) 2 F 4 + (2α ) 3( F 5 + D 2 F 4) + +(2α ) 4( F 6 + D 2 F 5 + D 4 F 4) + O ( (2α ) 4) ] 113
114 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. 114
115 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. ii) The construction of the (tree level) N-point amplitude of gauge bosons, A(1, 2,..., N). 115
116 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. ii) The construction of the (tree level) N-point amplitude of gauge bosons, A(1, 2,..., N). A(1, 2,..., N) = F (N) 1 A Y M (1, 2,..., N 1, N) F (N) (N 3)! A Y M(1, N 2,..., N 1, N), 116
117 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. ii) The construction of the (tree level) N-point amplitude of gauge bosons, A(1, 2,..., N). 117
118 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. ii) The construction of the (tree level) N-point amplitude of gauge bosons, A(1, 2,..., N). This is acheived by using 118
119 1. Our revisited S-matrix approach, which deals with the RNS description of the open superstring, proposes a CONCRETE SHORTCUT to: i) The construction of the LEL, L e. ii) The construction of the (tree level) N-point amplitude of gauge bosons, A(1, 2,..., N). This is acheived by using I. The kinematical constraints (absence of (ζ k) N terms) + II. The Open Superstring 4-point amplitude 119
120 2. Although at this moment, 120
121 2. Although at this moment, neither L e 121
122 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), 122
123 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, 123
124 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, 124
125 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, bypassing the computation of α expansion of worldsheet integrals that arise in N 5 point amplitudes, 125
126 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, bypassing the computation of α expansion of worldsheet integrals that arise in N 5 point amplitudes, like, for example, for N=6 : 1 0 x4 x3 2α dx 4 dx 3 dx 2 x α 12 2α 2 x α 13 2α 3 x α (x 3 x 2 ) 2α α (x 4 x 2 ) 2α α 24 (x 4 x 3 ) 2α α 34 1 (1 x 2 ) 2α α 25 1 (1 x 3 ) 2α α 35 (1 x 4 ) 2α α
127 1 0 x4 x3 2α dx 4 dx 3 dx 2 x α 12 2α 2 x α 13 2α 3 x α (x 3 x 2 ) 2α α (x 4 x 2 ) 2α α 24 (x 4 x 3 ) 2α α 34 1 (1 x 2 ) 2α α 25 1 (1 x 3 ) 2α α 35 (1 x 4 ) 2α α 45 [( (2α ) 3 α 23 α 16 t ζ(2) [ ( α16 + α 56 (2α ) 1 α 23 t O((2α ) 0 ) ( 1 + α α 34 = ) ( )] α 34 α 56 t 234 α 34 α 16 t 234 α 23 α 56 t 234 ) ( ) ( α23 + α ) 34 + t 234 α 34 α 56 α 16 t 234 α 56 t 234 α ) ( 16 α23 + α ) ( 34 α12 + α ) 45 t 234 α 16 t 234 α 56 α 56 α 34 α 23 α ( 16 α56 + α ) ( 16 t345 + t )] α 23 t 234 t 234 α 34 α 34 α 16 α 23 α 56 where α ij = k i k j and t mnp = α mn + α mp + α np. 127
128 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, bypassing the computation of α expansion of worldsheet integrals that arise in N 5 point amplitudes. 128
129 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, bypassing the computation of α expansion of worldsheet integrals that arise in N 5 point amplitudes. 3. Our revisited S-matrix method, also has a version for the fermionic terms of L e. 129
130 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, bypassing the computation of α expansion of worldsheet integrals that arise in N 5 point amplitudes. 3. Our revisited S-matrix method, also has a version for the fermionic terms of L e. This result will appear on a forthcoming work. 130
131 2. Although at this moment, neither L e nor the momentum factors F (N) j, of A(1, 2,..., N), are known at any α order, our revisited S-matrix method suggests an explicit construction of both of them, bypassing the computation of α expansion of worldsheet integrals that arise in N 5 point amplitudes. 3. Our revisited S-matrix method, also has a version for the fermionic terms of L e. This result will appear on a forthcoming work. Obrigado! 131
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